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Thread: Continuity

  1. #1
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    Continuity

    Hey.

    Could someone help me with the following?

    A function f on R to R satisfies for all x,y
    abs(f(x)-f(y)) less than or equal to k*abs(x-y) where k is a constant greater than zero.

    First, prove that f is continuous. Also, use this to show that given an epsilon greater than 0, there is at least one delta greater than zero that "measures" the continuity for this epsilon at all points.

    Thanks a lot. I have no idea what I'm doing.
    Last edited by JackStolerman; Nov 10th 2006 at 09:50 AM. Reason: Mistake
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  2. #2
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    Quote Originally Posted by JackStolerman View Post
    A function f on R to R satisfies for all x,y
    abs(f(x)-f(y)) less than or equal to k*abs(x-y) where k is a constant greater than zero.
    First, prove that f is continuous.
    Of course, this is a Lipschitz condition. It not only implies continuity, but uniform continuity!
    It works for any K > 0.
    To see that: given \varepsilon  > 0 choose \delta  = \frac{\varepsilon }{K}.

    Quote Originally Posted by JackStolerman View Post
    show that given an epsilon greater than 0, there is at least one delta greater than zero that "measures" the continuity for this epsilon at all points.
    I am not sure what that means. K is sometimes called the Lipschitz Constant.
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